Seems like the Generate Near Table tool would be the place to start. You would set the Maximum number of closest matches parameter to 2. This will yield a table with two records for each building. It won't have the road width on the table, but you could use something like the Join Field tool to add this information to the table. The next step is to pivot the table. That is, you don't want two records per building, you want one record with fields like:
Road1ID (closest road ID)
Road2ID (second closest road)
Off the top of my head, I'd look at the Pivot Table tool to do this, but I'm not sure if it will do exactly what you need. I'll have to think more about it over lunch .